Question: $\dfrac{ -3v + 4w }{ -4 } = \dfrac{ 9v + x }{ 6 }$ Solve for $v$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -3v + 4w }{ -{4} } = \dfrac{ 9v + x }{ 6 }$ $-{4} \cdot \dfrac{ -3v + 4w }{ -{4} } = -{4} \cdot \dfrac{ 9v + x }{ 6 }$ $-3v + 4w = -{4} \cdot \dfrac { 9v + x }{ 6 }$ Multiply both sides by the right denominator. $-3v + 4w = -4 \cdot \dfrac{ 9v + x }{ {6} }$ ${6} \cdot \left( -3v + 4w \right) = {6} \cdot -4 \cdot \dfrac{ 9v + x }{ {6} }$ ${6} \cdot \left( -3v + 4w \right) = -4 \cdot \left( 9v + x \right)$ Distribute both sides ${6} \cdot \left( -3v + 4w \right) = -{4} \cdot \left( 9v + x \right)$ $-{18}v + {24}w = -{36}v - {4}x$ Combine $v$ terms on the left. $-{18v} + 24w = -{36v} - 4x$ ${18v} + 24w = -4x$ Move the $w$ term to the right. $18v + {24w} = -4x$ $18v = -4x - {24w}$ Isolate $v$ by dividing both sides by its coefficient. ${18}v = -4x - 24w$ $v = \dfrac{ -4x - 24w }{ {18} }$ All of these terms are divisible by $2$ $v = \dfrac{ -{2}x - {12}w }{ {9} }$